Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 954684, 15 pages
doi:10.1155/2010/954684
Research Article
Dynamical Analysis of a Delayed Predator-Prey
System with Birth Pulse and Impulsive Harvesting
at Different Moments
Jianjun Jiao1 and Lansun Chen2
1
Guizhou Key Laboratory of Economic System Simulation, School of Mathematics and Statistics,
Guizhou College of Finance and Economics, Guiyang 550004, China
2
Institute of Mathematics, Academy of Mathematics and System Sciences, Beijing 100080, China
Correspondence should be addressed to Jianjun Jiao, jiaojianjun05@126.com
Received 21 August 2010; Accepted 22 September 2010
Academic Editor: Kanishka Perera
Copyright q 2010 J. Jiao and L. Chen. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider a delayed Holling type II predator-prey system with birth pulse and impulsive
harvesting on predator population at different moments. Firstly, we prove that all solutions of the
investigated system are uniformly ultimately bounded. Secondly, the conditions of the globally
attractive prey-extinction boundary periodic solution of the investigated system are obtained.
Finally, the permanence of the investigated system is also obtained. Our results provide reliable
tactic basis for the practical biological economics management.
1. Introduction
Theories of impulsive differential equations have been introduced into population dynamics
lately 1, 2. Impulsive equations are found in almost every domain of applied science and
have been studied in many investigation 3–11, they generally describe phenomena which
are subject to steep or instantaneous changes. In 11, Jiao et al. suggested releasing pesticides
is combined with transmitting infective pests into an SI model. This may be accomplished
using selecting pesticides and timing the application to avoid periods when the infective
pesticides would be exposed or placing the pesticides in a location where the transmitting
infective pests would not contact it. So an impulsive differential equation to model the process
of releasing infective pests and spraying pesticides at different fixed moment was represented
as
St θIt
dSt
rSt 1 −
− βStIt,
dt
K
t
/ n − 1 lτ, t / nτ,
dIt
βStIt − It,
dt
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ΔSt −μ1 St,
ΔIt −μ2 It,
ΔSt 0,
ΔIt μ,
t n − 1 lτ, n 1, 2, . . . ,
t nτ, n 1, 2, . . . .
1.1
The biological meaning of the parameters in System 1.1 can refer to Literature 11.
Clack 12 has studied the optimal harvesting of the logistic equation, a logistic
equation without exploitation as follows:
dxt
xt
rxt 1 −
,
dt
K
1.2
where xt represents the density of the resource population at time t, r is the intrinsic growth
rate. the positive constant K is usually referred as the environmental carrying capacity or
saturation level. Suppose that the population described by logistic equation 1.1 is subject to
harvesting at rate ht constant or under the catch-per-unit effort hypothesis ht Ext.
Then the equations of the harvested population revise, respectively, as following
xt
dxt
rxt 1 −
− h,
dt
K
1.3
dxt
xt
rxt 1 −
− Ext,
dt
K
1.4
or
where E denotes the harvesting effort.
Moreover, in most models of population dynamics, increase in population due to birth
are assumed to be time dependent, but many species reproduce only during a period of the
year. In between these pulses of growth, mortality takes its toll, and the population decreases.
In this paper, we suggest impulsive differential equations to model the process of periodic
birth pulse and impulsive harvesting. Combining 1.2 and 1.4, we can obtain a single
population model with birth pulse and impulsive harvesting at different moments
dxt
−dxt,
dt
t
/ n lτ, t / n 1τ,
Δxt xta − bxt,
Δxt −μxt,
t n lτ,
1.5
t n 1τ, n ∈ Z ,
where xt is the density of the population. d is the death rate. The population is birth pulse
as intrinsic rate of natural increase and density dependence rate of predator population are
denoted by a, b, respectively. The pulse birth and impulsive harvesting occurs every τ period
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τ is a positive constant. Δxt xt − xt. xta − bxt represents the birth effort of
predator population at t n lτ, 0 < l < 1, n ∈ Z . 0 ≤ μ ≤ 1 represents the harvesting
effort of predator population at t n 1τ, n ∈ Z .
But in the natural world, there are many species especially insects whose individual
members have a life history that takes them through two stages, immature and mature. In
13, a stage-structured model of population growth consisting of immature and mature
individuals was analyzed, where the stage-structured was modeled by introduction of a
constant time delay. Other models of population growth with time delays were considered in
3, 5–7, 13. The following single- species stage-structured model was introduced by Aiello
and Freedman 14 as follows:
x t βyt − rxt − βe−rτ yt − τ,
y t βe−rτ yt − τ − η2 y2 t,
1.6
where xt, yt represent the immature and mature populations densities, respectively, τ
represents a constant time to maturity, and β, r and η2 are positive constants. This model is
derived as follows. We assume that at any time t > 0, birth into the immature population
is proportional to the existing mature population with proportionality constant β. We then
assume that the death rate of immature population is proportional to the existing immature
population with proportionality constant r. We also assume that the death rate of mature
population is of a logistic nature, that is, proportional to the square of the population with
proportionality constant η2 . In this paper, we consider a delayed Holling type II predatorprey system with birth pulse and impulsive harvesting on predator population at different
moments.
The organization of this paper is as follows. In the next section, we introduce the
model. In Section 3, some important lemmas are presented. In Section 4, we give the globally
asymptotically stable conditions of prey-extinction periodic solution of System 2.1, and the
permanent condition of System 2.1. In Section 5, a brief discussion is given in the last section
to conclude this paper.
2. The Model
In this paper, we consider a delayed Holling type II predator-prey model with birth pulse
and impulsive harvesting on predator population at different moments
dx1 t
rx2 t − re−wτ1 x2 t − τ1 − wx1 t,
dt
βx2 t
dx2 t
re−wτ1 x2 t − τ1 −
yt − d1 x2 t,
dt
m x2 t
dyt
kβx2 t
yt − d2 yt,
dt
m x2 t
Δx1 t 0,
Δx2 t 0,
Δyt yt a − byt ,
t n lτ, n 1, 2 . . . ,
t/
n lτ, t /
n 1τ,
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Δx1 t 0,
Δx2 t 0,
t n 1τ, n 1, 2 . . . ,
Δyt −μyt,
2.1
the initial conditions for 2.1 are
ϕ1 ζ, ϕ2 ζ, ϕ3 ζ ∈ C C −τ1 , 0, R3 ,
ϕi 0 > 0,
i 1, 2, 3,
2.2
where x1 t, x2 t represent the densities of the immature and mature prey populations,
respectively. yt represents the density of predator population. r > 0 is the intrinsic growth
rate of prey population. τ1 represents a constant time to maturity. w is the natural death rate
of the immature prey population. d1 is the natural death rate of the mature prey population.
d2 is the natural death rate of the predator population. The predator population consumes
prey population following a Holling type-II functional response with predation coefficients
β, and half-saturation constant m. k is the rate of conversion of nutrients into the reproduction
rate of the predators. The predator population is birth pulse as intrinsic rate of natural
increase and density dependence rate of predator population are denoted by a, b, respectively.
The pulse birth and impulsive harvesting occurs every τ period τ is a positive constant.
Δyt yt − yt. yta − byt represents the birth effort of predator population at
t n lτ, 0 < l < 1, n ∈ Z . 0 ≤ μ ≤ 1 represents the harvesting effort of predator population
at t n 1τ, n ∈ Z . In this paper, we always assume that τ < 1/d ln1 a.
Before going into any details, we simplify model 2.1 and restrict our attention to the
following model:
βx2 t
dx2 t
re−wτ1 x2 t − τ1 −
yt − d1 x2 t,
dt
m x2 t
dyt
kβx2 t
yt − d2 yt,
dt
m x2 t
Δx2 t 0,
Δyt yt a − byt ,
Δx2 t 0,
Δyt −μyt,
t/
n lτ, t /
n 1τ,
2.3
t n lτ, n 1, 2, . . . ,
t n 1τ, n 1, 2, . . . ,
the initial conditions for 2.3 are
ϕ2 ζ, ϕ3 ζ ∈ C C −τ1 , 0, R2 ,
ϕi 0 > 0,
i 2, 3.
2.4
3. The Lemma
Before discussing main results, we will give some definitions, notations and lemmas. Let
R 0, ∞, R3 {x ∈ R3 : x > 0}. Denote f f1 , f2 , f3 the map defined by the right hand
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of system 2.1. Let V : R × R3 → R , then V is said to belong to class V0 , if
i V is continuous in nτ, n lτ × R3 and n lτ, n 1τ × R3 , for each x ∈ R3 ,
n ∈ Z , limt,y → nlτ ,x V t, y V n lτ , x and limt,y → n1τ ,x V t, y V n 1τ , x exist.
ii V is locally Lipschitzian in x.
Definition 3.1. V ∈ V0 , then for t, z ∈ nτ, n lτ × R3 and n lτ, n 1τ × R3 , the upper
right derivative of V t, z with respect to the impulsive differential system 2.1 is defined as
D V t, z lim sup
h→0
1 V t h, z hft, z − V t, z .
h
3.1
The solution of 2.1, denote by zt xt, ytT , is a piecewise continuous function x:R →
R3 , zt is continuous on nτ, n lτ × R3 and n lτ, n 1τ × R3 n ∈ Z , 0 ≤ l ≤ 1.
Obviously, the global existence and uniqueness of solutions of 2.1 is guaranteed by the
smoothness properties of f, which denotes the mapping defined by right-side of system 2.1
Lakshmikantham et al. 1. Before we have the the main results. we need give some lemmas
which will be used as follows.
Now, we show that all solutions of 2.1 are uniformly ultimately bounded.
Lemma 3.2. There exists a constant M > 0 such that x1 t ≤ M/k, x2 t ≤ M/k, yt ≤ M for
each solution x1 t, x2 t, yt of 2.1 with all t large enough.
Proof. Define V t kx1 t kx2 t yt.
nτ, we have
i If d1 > r, then d min{d1 , d2 , d1 − r}, when t /
Δ
D V t dV t −kd1 − r − dx1 t − kd2 − dx2 t − d2 − dyt ξ ≤ 0.
3.2
When t n l − 1τ,
V n lτ kxn lτ yn lτ yn lτ a − byn lτ
a 2 a2
V n lτ − b yn lτ −
2b
4b
≤ V n lτ 3.3
a2
.
4b
Δ
For convenience, we make a notation as ξ1 a2 /4b. When t nτ,
V n 1τ kxn 1τ 1 − μ yn 1τ ≤ V n 1τ.
3.4
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From 17, Lemma 2.2, Page 23 for t ∈ n − 1τ, n l − 1τ and n l − 1τ, nτ, we have
V t ≤ V 0 e−dt e−dt−τ
edτ
edτ
ξ
ξ
1 − e−dt ξ1
ξ
−→
,
ξ
1
1
d
d
1 − e−dτ
edτ − 1
edτ − 1
as t −→ ∞.
3.5
ii If d1 < r, then d 0, we can easily obtain
V t ≤ V 0 ,
as t −→ ∞.
3.6
So V t is uniformly ultimately bounded. Hence, by the definition of V t, there exists a
constant M > 0 such that xt ≤ M/k, yt ≤ M for t large enough. The proof is complete.
If xt 0, we have the following subsystem of System 2.1:
dyt
−d2 yt, t /
n lτ, t /
n 1τ,
dt
Δyt yt a − byt , t n lτ,
Δyt −μyt,
3.7
t n 1τ, n ∈ Z .
We can easily obtain the analytic solution of System 3.7 between pulses, that is,
⎧
⎨ynτ e−d2 t−nτ ,
yt ⎩ 1 ae−d2 lτ ynτ be−2d2 lτ y2 nτ e−d2 t−nlτ ,
t ∈ nτ, n lτ,
t ∈ n lτ, n 1τ.
3.8
Considering the last two equations of system 3.7, we have the stroboscopic map of System
3.7 as follows:
yn 1τ 1 − μ 1 ae−d2 τ ynτ − 1 − μ be−d2 1lτ y2 nτ .
3.9
The are two fixed points of 3.9 are obtained as G1 0 and G2 y∗ , where
y∗ 1 a d2 lτ
1
e
−
ed2 1lτ
b
1−μ b
with μ < 1 −
1 d2 τ
e .
1a
3.10
Lemma 3.3. i If μ > 1 − 1/1 aed2 τ , the fixed point G1 0 is globally asymptotically stable;
ii if μ < 1 − 1/1 aed2 τ , the fixed point G2 y∗ is globally asymptotically stable.
Proof. For convenience, make notation yn ynτ , then Difference equation 3.9 can be
rewritten as
yn1 F yn .
3.11
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i If μ > 1 − 1/1 aed2 τ , G1 0 is the unique fixed point, we have
dFy 1 − μ 1 ae−d2 τ < 1,
dy y0
3.12
then G1 0 is globally asymptotically stable.
ii If μ < 1 − 1/1 aed2 τ , G1 0 is unstable. For
dFy − 1 − μ 1 ae−d2 τ 2 < 1,
dy yy∗
3.13
then G1 y∗ is globally asymptotically stable. This complete the proof.
It is well known that the following lemma can easily be proved.
Lemma 3.4. i If μ > 1 − 1/1 aed2 τ , the triviality periodic solution of System 3.7 is globally
asymptotically stable;
ii if μ < 1 − 1/1 aed2 τ , the periodic solution of System 3.7
⎧
⎨y∗ e−d2 t−nτ ,
yt ⎩ 1 ae−d2 lτ y∗ be−2d2 lτ y∗ 2 e−d2 t−nlτ ,
t ∈ nτ, n lτ,
t ∈ n lτ, n 1τ
3.14
is globally asymptotically stable. Here,
y∗ 1 a d2 lτ
1
e
−
ed2 1lτ .
b
1−μ b
3.15
Lemma 3.5 see 22. Consider the following delay equation:
x t a1 xt − τ − a2 xt 0,
3.16
one assumes that a1 , a2 , τ > 0; xt > 0 for −τ ≤ t ≤ 0. Assume that a1 < a2 . Then
lim xt 0.
t→∞
3.17
4. The Dynamics
In this section, we will firstly obtain the sufficient condition of the global attractivity of preyextinction periodic solution of System 2.1 with 2.2.
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Theorem 4.1. If
1 d2 τ
e ,
1a
2 d1
1 ae−d2 τ y∗ be−d2 1lτ y∗
μ<1−
re−wτ1 <
kβ −d2 lτ
e
km M
4.1
4.2
of System 2.1 with 2.2 is globally attractive
hold, the prey-extinction solution 0, 0, yt
y∗ 1 a d2 lτ
1
e
−
ed2 1lτ .
b
1−μ b
4.3
Proof. It is clear that the global attraction of prey-extinction periodic solution 0, 0, yt
of System 2.1 with 2.2 is equivalent to the global attraction of prey-extinction periodic
of System 2.3. So we only devote to System 2.3 with 2.4. Since
solution 0, yt
re−wτ1 <
2 kβ −d2 lτ
e
d1 ,
1 ae−d2 τ y∗ be−d2 1lτ y∗
km M
4.4
we can choose ε0 sufficiently small such that
re−wτ1 <
2
kβ −d2 lτ
e
1 ae−d2 τ y∗ be−d2 1lτ y∗ − ε0 d1 .
km M
4.5
It follows from that the second equation of System 2.3 with 2.4 that dyt/dt ≥ −d2 yt.
So we consider the following comparison impulsive differential system:
dxt
−d2 xt,
dt
t/
n lτ, t /
n 1τ,
Δxt xta − bxt,
Δxt −μxt,
t n lτ,
4.6
t n 1τ.
In view of Condition 4.1 and Lemma 3.4, we obtain that the periodic solution of System
4.6
⎧
⎨x∗ e−d2 t−nτ ,
t ∈ nτ, n lτ,
xt ⎩ 1 ae−d2 lτ x∗ be−2d2 lτ x∗ 2 e−d2 t−nlτ , t ∈ n lτ, n 1τ,
4.7
is globally asymptotically stable. Here,
x∗ 1 a d2 lτ
1
e
−
ed2 1lτ .
b
1−μ b
4.8
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By the comparison theorem of impulsive equation see 13, Theorem 3.1.1, we have
yt
as t → ∞. Then there exists an integer k2 > k1 , t > k2 such
yt ≥ xt and xt → xt
that
− ε0 ,
yt ≥ xt ≥ yt
nτ < t ≤ n 1τ,
4.9
n > k2 ,
that is
− ε0 ≥
yt > yt
2 Δ
− ε0 ,
e−d2 lτ 1 ae−d2 τ y∗ be−d2 1lτ y∗
nτ < t ≤ n 1τ,
4.10
n > k2 .
From 2.3, we get
dx2 t
≤ re−wτ1 x2 t − τ1 −
dt
kβ
d1 x2 t,
km M
t > nτ τ1 ,
n > k2 .
4.11
Consider the following comparison differential system:
dzt
re−wτ1 zt − τ1 −
dt
kβ
d1 zt,
km M
t > nτ τ1 ,
n > k2 ,
4.12
from 4.5, we have re−wτ1 < kβ
/km M d1 . According to Lemma 3.5, we have
limt → ∞ zt 0.
Let x2 t, yt be the solution of system 2.3 with initial conditions 2.4 and x2 ζ ϕ2 ζ ζ ∈ −τ1 , 0, yt is the solution of system 4.12 with initial conditions zζ ϕ2 ζζ ∈
−τ1 , 0. By the comparison theorem, we have limt → ∞ x2 t < limt → ∞ zt 0. Incorporating
into the positivity of x2 t, we know that
lim x2 t 0.
t→∞
4.13
Therefore, for any ε1 > 0 sufficiently small, there exists an integer k3 k3 τ > k2 τ τ1 such
that x2 t < ε1 for all t > k3 τ.
For System 2.3, we have
dyt
≤
−d2 yt ≤
dt
kβε1
−d2 yt,
m ε1
4.14
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z2 t → yt
as t → ∞, while z1 t and
then we have z1 t ≤ yt ≤ z2 t and z1 t → yt,
z2 t are the solutions of
dz1 t
−d2 z1 t,
dt
t/
n lτ, t /
n 1τ,
Δz1 t z1 ta bz1 t,
Δz1 t −μz1 t,
dz2 t
dt
kβε1
−d2 z2 t,
m ε1
t n 1τ,
4.15
t/
n lτ, t /
n 1τ,
Δz2 t z2 ta bz2 t,
Δz2 t −μz2 t,
t n lτ,
t n lτ,
t n 1τ,
respectively,
⎧ ∗ −d kβε /mε t−nτ
1
1
,
t ∈ nτ, n lτ,
ze 2
⎪
⎪
⎪ 2
⎨
2
z
1 ae−d2 kβε1 /mε1 lτ z∗2 be2−d2 kβε1 /mε1 lτ z∗2
2 t ⎪
⎪
⎪
⎩
×e−d2 kβε1 /mε1 t−nlτ ,
t ∈ n lτ, n 1τ,
4.16
Here,
z∗2 1 a d2 −kβε1 /mε1 lτ
1
e
−
ed2 −kβε1 /mε1 1lτ .
b
1−μ b
4.17
Therefore, for any ε2 > 0. there exists a integer k4 , n > k4 such that
ε2 ,
− ε2 < yt < yt
yt
4.18
ε2 ,
− ε2 < yt < yt
yt
4.19
Let ε1 → 0, so we have
as t → ∞. This completes the proof.
for t large enough, which implies yt → yt
The next work is to investigate the permanence of the system 2.4. Before starting our
theorem, we give the definition of permanence of system 2.4.
Definition 4.2. System 2.1 is said to be permanent if there are constants m, M > 0
independent of initial value and a finite time T0 such that for all solutions x1 t, x2 t, yt
with all initial values x1 t > 0, x2 0 > 0, y0 > 0, m ≤ x1 t < M/k, x2 t ≤ M/k, m ≤
x3 t ≤ M holds for all t ≥ T0 . Here T0 may depend on the initial values x1 0 , x2 0 , y0 .
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Theorem 4.3. If
re−wτ1 >
β ∗
∗
∗
∗
1 1 ae−d2 kβx2 /mx2 lτ v∗ be2−d2 kβx2 /mx2 lτ v∗ 2 d1 ,
m
4.20
then there is a positive constant q such that each positive solution x2 t, yt of 2.3 with 2.4
satisfies
x2 t ≥ q,
4.21
for t large enough, where x2∗ is determined as the following equation:
1 1 ae
−d2 kβx2∗ /mx2∗ lτ
1 a d2 −kβx2∗ /mx2∗ lτ
1
∗
∗
×
e
−
ed2 −kβx2 /mx2 1lτ
b
1−μ b
be
2−d2 kβx2∗ /mx2∗ lτ
×
1
1 a d2 −kβx2∗ /mx2∗ lτ
∗
∗
e
−
ed2 −kβx2 /mx2 1lτ
b
1−μ b
m −wτ1
− d1 .
re
β
2
4.22
Proof. The first equation of System 2.3 can be rewritten as
dx2 t
dt
re−wτ1 −
βyt
d t
− d1 x2 t − re−wτ1
x2 udu.
m x2 t
dt t−τ1
4.23
Let us consider any positive solution x2 t, yt of System 2.3. According to4.23, V t is
defined as
V t x2 t re
−wτ1
t
x2 udu.
4.24
t−τ1
We calculate the derivative of V t along the solution of 2.3 as follows:
βyt
dV t
re−wτ1 −
− d1 x2 t,
dt
m x2 t
4.25
Equation 4.25 can also be written
β
dV t
−wτ1
> re
− yt − d1 x2 t.
dt
m
4.26
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We claim that for any t0 > 0, it is impossible that x2 t < x2∗ for all t > t0 . Suppose that
the claim is not valid. Then there is a t0 > 0 such that x2 t < x2∗ for all t > t0 . It follows from
the second equation of System 2.3 that for all t > t0 ,
dyt
<
dt
kβx2∗
− d2 yt.
m x2∗
4.27
Consider the following comparison impulsive system for all t > t0
dvt
dt
kβx2∗
−
d
2 vt,
m x2∗
t
/ n lτ, n 1τ,
Δvt vta − bvt,
Δvt −μvt,
4.28
t n lτ,
t n 1τ.
By Lemma 3.4, we obtain
⎧ ∗ −d kβx∗ /mx∗ t−nτ
2
2
,
t ∈ nτ, n lτ,
v e 2
⎪
⎪
⎪
⎨
∗
∗
∗
∗
vt
1 ae−d2 kβx2 /mx2 lτ v∗ be2−d2 kβx2 /mx2 lτ v∗ 2
⎪
⎪
⎪
∗
∗
⎩
×e−d2 kβx2 /mx2 t−nlτ ,
t ∈ n lτ, n 1τ,
4.29
is the unique positive periodic solution of 4.28 which is globally asymptotically stable,
where
v∗ 1 a d2 −kβx2∗ /mx2∗ lτ
1
∗
∗
e
−
ed2 −kβx2 /mx2 1lτ .
b
1−μ b
4.30
By the comparison theorem for impulsive differential equation 1, 2, we know that there
exists t1 > t0 τ1 such that the following inequality holds for t ≥ t1 :
ε.
yt ≤ vt
4.31
∗
∗
∗
∗
yt ≤ 1 1 ae−d2 kβx2 /mx2 lτ v∗ be2−d2 kβx2 /mx2 lτ v∗ 2 ε,
4.32
Thus,
∗
∗
for all t ≥ t1 . For convenience, we make notation as σ 1 1 ae−d2 kβx2 /mx2 lτ v∗ ∗
∗
be2−d2 kβx2 /mx2 lτ v∗ 2 ε. From 4.20, we can choose a ε such that have
re−wτ1 >
β
∗
∗
∗
∗
1 1 ae−d2 kβx2 /mx2 lτ v∗ be2−d2 kβx2 /mx2 lτ v∗ 2 ε d1 ,
m
4.33
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By 4.26, we have
β
V t > x2 t re−wτ1 − σ − d1 ,
m
4.34
for all t > t1 . Set
x2m min x2 t,
t∈t1 ,t1 τ1 4.35
We will show that x2 t ≥ x2m for all t ≥ t1 . Suppose the contrary. Then there is a T0 > 0 such
that x2 t ≥ x2m for t1 ≤ t ≤ t1 τ1 T0 , x2 t1 τ1 T0 x2m and x2 t1 τ1 T0 < 0. Hence, the
first equation of system 2.3 and 4.33 imply that
x2 t1 τ1 T0 re−wτ1 x2 t1 T0 −
βx2 t1 τ1 T0 yt1 τ1 T0 m x2 t1 τ1 T0 − d1 x2 t1 τ1 T0 ,
β
≥ re−wτ1 − σ − d1 x2m
m
4.36
> 0.
This is a contradiction. Thus, x2 t ≥ x2m for all t > t1 . As a consequence, 4.26 and 4.33 lead
to
β
V t > x2m re−wτ1 − σ − d1 > 0,
m
4.37
for all t > t1 . This implies that as t → ∞, V t → ∞. It is a contradiction to V t ≤ M1 rτ1 e−wτ1 . Hence, the claim is complete.
By the claim, we are left to consider two case. First, x2 t ≥ x2∗ for all t large enough.
Second, x2 t oscillates about x2∗ for t large enough.
Define
q min
x2∗
, q1 ,
2
4.38
where q1 x2∗ e−βM/mMd1 τ1 . We hope to show that x2 t ≥ q for all t large enough. The
conclusion is evident in first case. For the second case, let t∗ > 0 and ξ > 0 satisfy x2 t∗ x2 t∗ ξ x2∗ and x2 t < x2∗ for all t∗ < t < t∗ ξ where t∗ is sufficiently large such that
yt < σ
for t∗ < t < t∗ ξ,
4.39
x2 t is uniformly continuous. The positive solutions of 2.3 are ultimately bounded and
x2 t is not affected by impulses. Hence, there is a T 0 < t < τ1 and T is dependent of the
14
Advances in Difference Equations
choice of t∗ such that x2 t∗ > x2∗ /2 for t∗ < t < t∗ T . If ξ < T , there is nothing to prove. Let
us consider the case T < ξ < τ1 . Since x2 t > −βM/m M d1 x2 t and x2 t∗ x2∗ , it
is clear that x2 t ≥ q1 for t ∈ t∗ , t∗ τ1 . Then, proceeding exactly as the proof for the above
claim. We see that x2 t ≥ q1 for t ∈ t∗ τ1 , t∗ ξ. Because the kind of interval t ∈ t∗ , t∗ ξ
is chosen in an arbitrary way we only need t∗ to be large. We concluded x2 t ≥ q for all
large t. In the second case. In view of our above discussion, the choice of q is independent of
the positive solution, and we proved that any positive solution of 2.3 satisfies x2 t ≥ q for
all sufficiently large t. This completes the proof of the theorem.
From Theorems 4.1 and 4.3, we can easily obtain the following theorem.
Theorem 4.4. If
re−wτ1 >
β ∗
∗
∗
∗
1 1 ae−d2 kβx2 /mx2 lτ v∗ be2−d2 kβx2 /mx2 lτ v∗ 2 d1 ,
m
4.40
then System 2.1 with 2.2 is permanent, where x2∗ is determined as the following equation:
1 a d2 −kβx∗ /mx∗ lτ
1
∗
∗
−d2 kβx2∗ /mx2∗ lτ
d
−kβx
/mx
1lτ
2
2
2
2
2
e
×
−
1 1 ae
e
b
1−μ b
∗
∗
be2−d2 kβx2 /mx2 lτ ×
m −wτ1
− d1 .
re
β
1
1 a d2 −kβx2∗ /mx2∗ lτ
∗
∗
e
ed2 −kβx2 /mx2 1lτ
−
b
1 − μb
2
4.41
5. Discussion
In this paper, considering the fact of the biological source management, we consider a delayed
Holling type II predator-prey system with birth pulse and impulsive harvesting on predator
population at different moments. We prove that all solutions of System 2.1 with 2.2
are uniformly ultimately bounded. The conditions of the globally attractive prey-extinction
boundary periodic solution of System 2.1 with 2.2 are obtained. The permanence of the
System 2.1 with 2.2 is also obtained. The results show that the successful management of
a renewable resource is based on the concept of a sustain yield, that is, an exploitation does
not the threaten the long-term persistence of the species. Our results provide reliable tactic
basis for the practical biological resource management.
Acknowledgments
This work was supported by National Natural Science Foundation of China no. 10961008,
the Nomarch Foundation of Guizhou Province no. 2008035, the Science Technology
Foundation of Guizhou Education Department no. 2008038, and the Science Technology
Foundation of Guizhou no. 2010J2130.
Advances in Difference Equations
15
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